Question

use the values $sin 45^circ = 0.707, cos 45^circ = 0.707, \tan 45^circ = 1$

1. determine the information that is given and what you are trying to find.

given: a. $mangle a = $
b. length of $b = $
find: hypotenuse $(c)$

1. choose the correct trig ratio.
2. solve for the unknown.
3. substitute the given information.
4. solve the problem.

options (partial): a. $60^circ$; b. $45^circ$; c. $hyp = \frac{5}{cos 45^circ}$ (note: original $b$ length is 5 cm?); d. $7.07$ cm; e. $cos a = \frac{adj}{hyp}$; f. $\tan$; g. $hyp = \frac{opp}{sin a}$; h. $hyp = \frac{adj}{cos a}$; i. $sin a = \frac{opp}{hyp}$

Explanation:

1. Determine the given and find

Step 1: Analyze angle \( \angle A \)

From the diagram, \( m\angle A = 45^\circ \) (option b).

Step 2: Analyze length of \( b \)

The length of \( b \) (adjacent side to \( \angle A \)) is \( 5 \) cm.

2. Choose the correct trig ratio

Step 1: Recall trigonometric ratios

For a right - triangle, \( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \). Here, \( \theta = \angle A=45^\circ \), adjacent side \( = b = 5 \) cm, and hypotenuse \( = c \). So the correct trig ratio is \( \cos A=\frac{\text{adj}}{\text{hyp}} \) (option e).

3. Solve for the unknown

Step 1: Derive the formula for hypotenuse

From \( \cos A=\frac{\text{adj}}{\text{hyp}} \), we can re - arrange it to \( \text{hyp}=\frac{\text{adj}}{\cos A} \). Substituting \( \text{adj} = 5 \) and \( A = 45^\circ \), we get \( \text{hyp}=\frac{5}{\cos45^\circ} \) (option c).

4. Substitute the given information

Step 1: Substitute values

We know that \( \cos45^\circ = 0.707 \) and \( \text{hyp}=\frac{5}{\cos45^\circ} \), so substituting \( \cos45^\circ = 0.707 \) into the formula, we have \( \text{hyp}=\frac{5}{0.707} \) (option h).

5. Solve the problem

Step 1: Calculate the value

\( \frac{5}{0.707}\approx7.07 \) cm (option d).

1.

a. \( m\angle A=\boldsymbol{45^\circ} \) (option b)
b. length of \( b=\boldsymbol{5} \) cm

2.

The correct trig ratio is \( \boldsymbol{\cos A=\frac{\text{adj}}{\text{hyp}}} \) (option e)

3.

The formula to solve for hypotenuse is \( \boldsymbol{\text{hyp}=\frac{5}{\cos45^\circ}} \) (option c)

4.

After substitution, we have \( \boldsymbol{\text{hyp}=\frac{5}{0.707}} \) (option h)

5.

The length of the hypotenuse is \( \boldsymbol{7.07} \) cm (option d)

Answer:

1. Determine the given and find

Step 1: Analyze angle \( \angle A \)

From the diagram, \( m\angle A = 45^\circ \) (option b).

Step 2: Analyze length of \( b \)

The length of \( b \) (adjacent side to \( \angle A \)) is \( 5 \) cm.

2. Choose the correct trig ratio

Step 1: Recall trigonometric ratios

For a right - triangle, \( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \). Here, \( \theta = \angle A=45^\circ \), adjacent side \( = b = 5 \) cm, and hypotenuse \( = c \). So the correct trig ratio is \( \cos A=\frac{\text{adj}}{\text{hyp}} \) (option e).

3. Solve for the unknown

Step 1: Derive the formula for hypotenuse

From \( \cos A=\frac{\text{adj}}{\text{hyp}} \), we can re - arrange it to \( \text{hyp}=\frac{\text{adj}}{\cos A} \). Substituting \( \text{adj} = 5 \) and \( A = 45^\circ \), we get \( \text{hyp}=\frac{5}{\cos45^\circ} \) (option c).

4. Substitute the given information

Step 1: Substitute values

We know that \( \cos45^\circ = 0.707 \) and \( \text{hyp}=\frac{5}{\cos45^\circ} \), so substituting \( \cos45^\circ = 0.707 \) into the formula, we have \( \text{hyp}=\frac{5}{0.707} \) (option h).

5. Solve the problem

Step 1: Calculate the value

\( \frac{5}{0.707}\approx7.07 \) cm (option d).

1.

a. \( m\angle A=\boldsymbol{45^\circ} \) (option b)
b. length of \( b=\boldsymbol{5} \) cm

2.

The correct trig ratio is \( \boldsymbol{\cos A=\frac{\text{adj}}{\text{hyp}}} \) (option e)

3.

The formula to solve for hypotenuse is \( \boldsymbol{\text{hyp}=\frac{5}{\cos45^\circ}} \) (option c)

4.

After substitution, we have \( \boldsymbol{\text{hyp}=\frac{5}{0.707}} \) (option h)

5.

The length of the hypotenuse is \( \boldsymbol{7.07} \) cm (option d)